Anti-concentration of random variables from zero-free regions

Abstract

This paper provides a connection between the concentration of a random variable and the distribution of the roots of its probability generating function. Let $X$ be a random variable taking values in $\{0,\dots ,n\}$ with $\mathbb{P}(X=0)\mathbb{P}(X=n)>0$ and with probability generating function $f_X$. We show that if all of the zeros $\zeta$ of $f_X$ satisfy $|\arg (\zeta )|\ge \delta$ and $R^{-1}\le |\zeta |\le R$ then \[ \operatorname{Var}(X) \geq c R^{-2π/δ}n, \] where $c>0$ is a absolute constant. We show that this result is sharp, up to the factor $2$ in the exponent of $R$. As a consequence, we are able to deduce a Littlewood--Offord type theorem for random variables that are not necessarily sums of i.i.d.\ random variables.

29 pages

References in corpus (1)

• Central limit theorems and the geometry of polynomials